BOUNDARY EDGE DOMINATION IN GRAPHS

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1 BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) , ISSN (o) /JOURNALS / BULLETIN Vol. 5(015), Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA LUKA ISSN (o), ISSN X (p) BOUNDARY EDGE DOMINATION IN GRAPHS P uttaswamy and Mohammed Alatif Abstract. Let G = (V, E) be a connected graph, A subset S of E(G) is called a boundary edge dominating set if every edge of E S is edge boundary dominated by some edge of S. The minimum taken over all boundary edge dominating sets of a graph G is called the boundary edge domination number of G and is denoted by γ b (G). In this paper we introduce the edge boundary domination in graph. Exact values of some standard graphs are obtained and some other interesting results are established. 1. Introduction and Definitions For graph-theoretical terminology and notations not defined here we follow Buck-ley [6], West [8] and Haynes et al.[7]. All graphs in this paper will be finite and undirected, without loops and multiple edges. As usual n = V and m = E denote the number of vertices and edges of a graph G, respectively. In general, we use X to denote the subgraph induced by the set of vertices X. N(v) and N[v] denote the open and closed neighbourhood of a vertex v, respectively. A set D of vertices in a graph G is a dominating set if every vertex in V D is adjacent to some vertex in D. The domination number γ(g) is the minimum cardinality of a dominating set of G. A line graph L(G) (also called an interchange graph or edge graph) of a simple graph G is obtained by associating a vertex with each edge of the graph and connecting two vertices with an edge if and only if the corresponding edges of G have a vertex in common. For terminology and notations not specifically defined here we refer reader to [11]. Let G = (V, E) be a simple graph with vertex set V (G) = {v 1, v,..., v n }. For 010 Mathematics Subject Classification. : 05C69 05C8. Key words and phrases. Boundary dominating set, Boundary edge dominating set, Boundary edge domination number. The Authors would like to thank the referees for their comments. Also special thanks to Dr. Anwar Alwardi for his help and comments in this research work. 197

2 198 P UT T ASW AMY AND MOHAMMED ALATIF i j, a vertex v i is a boundary vertex of v j if d(v j ; v t ) d(v j ; v i ) for all v t N(v i ) (see [, 4]). A vertex v is called a boundary neighbor of u if v is a nearest boundary of u. If u V, then the boundary neighbourhood of u denoted by N b (u) is defined as N b (u) = {v V : d(u, w) d(u, v) for all w N(u)}. The cardinality of N b (u) is denoted by deg b (u) in G. The maximum and minimum boundary degree of a vertex in G are denoted respectively by b (G) and δ b (G). That is b (G) = max u V N b (u), δ b (G) = min u V N b (u). A vertex u boundary dominate a vertex v if v is a boundary neighbor of u. A subset B of V(G) is called a boundary dominating set if every vertex of V B is boundary dominated by some vertex of B. The minimum taken over all boundary dominating sets of a graph G is called the boundary domination number of G and is denoted by γ b (G). [] The distance d(e i, e j ) between two edges in E(G) is defined as the distance between the corresponding vertices e i and e j in the line graph of G, or if e i = uv and e j = u v, the distance between e i and e j in G is defined as follows: d(e i, e j ) = min{d(u, u ), d(u, v ), d(v, v ), d(v, u )}. The degree of an edge e = uv of G is defined by deg(e) = degu + degv. The concept of edge domination was introduced by Mitchell and Hedetniemi [1]. A subset X of E is called an edge dominating set of G if every edge not in X is adjacent to some edge in X. The edge domination number γ (G) of G is the minimum cardinality taken over all edge dominating sets of G. We need the following theorems. Theorem 1.1. [] a. For any path P n, n, γ b (P n ) = n. b. For any complete graph K n, n 4, γ b (K n ) = 1. Theorem 1.. [10] For any (n, m)-graph G, γ (G) m (G), where (G) denotes the maximum degree of an edge in G. Theorem 1.. [5] Edge - eccentric graph of a complete graph K n is a regular graph with regularity (n )(n ). Theorem 1.4. [1] If diam(g) and if none of the three graphs F 1, F, and F depicted in Fig. are induced subgraphs of G, then diam(l(g)). Figure 1. The graphs mentioned in Theorem 1.4

3 BOUNDARY EDGE DOMINATION IN GRAPHS 199. Results Let G = (V,E) be a graph and f, e be any two edges in E. Then f and e are adjacent if they have one end vertex in common. Definition.1. An edge e = uv E is said to be a boundary edge of f if d(e, g) d(e, f) for all g N (e). An edge g is a boundary neighbor of an edge f if g is a nearest boundary of f. Two edges f and e are boundary adjacent if f adjacent to e and there exist another edge g adjacent to both f and e. Definition.. A set S of edges is called a boundary edge dominating set if every edge of E S is boundary edge dominated by some edge of S. The minimum taken over all edge boundary dominating sets of a graph G is called the boundary edge domination number of G and is denoted by γ b (G). The boundary edge neighbourhood of e denoted by N b (e) is defined as N b (e) = {f E(G) : d(e, g) d(e, f) for all g N (e)}. The cardinality of N b (e) is denoted by deg b (e) in G. The maximum and minimum boundary degree of an edge in G are denoted respectively by b (G) and δ b (G). That is b (G) = max e E N b (e) and δ b (G) = min e E N b (e). A boundary edge dominating set S is minimal if for any edge f S, S {f} is not boundary edge dominating set of G. A subset S of E is called boundary edge independent set, if for any f S, f N b (g), for all g S {f}. If an edge f E be such that N b (f) = φ then j is in any boundary edge dominating set. Such edges are called boundary-isolated. The minimum boundary edge dominating set denoted by γ b (G)-set. An edge dominating set X is called an independent boundary edge dominating set if no two edges in X are boundary-adjacent. The independent boundary edge domination number γ bi (G) is the minimum cardinality taken over all independent boundary edge dominating sets of G. For a real number x; x denotes the greatest integer less than or equal to x and x denotes the smallest integer greater than or equal to x. In Figure, V (G) = {v 1, v, v, v 4, v 5, v 6 }, and E(G) = {1,,, 4, 5, 6}. The minimum boundary dominating set is B = {v, v }. Therefore γ b (G) =. The minimal edge dominating sets are {, 5}, {, 6}, {4, 6}, {1, 4}, {1,, 5}. Therefore γ (G) =. The minimum boundary edge dominating sets are {1,, 6}, {1, 5, 4}, {,, 4}, {4, 5, 6}. Therefore γ b (G) =. From the definition of line graph and the boundary edge domination the following Proposition is immediate. Observation 1. For any graph G, we have, γ b (G) = γ b(l(g)) The boundary edge domination number of some standard graphs are given below.

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